f@istortionforfffuftifa-Lorfectusionsjoin.twith Marcel Bischoff

Sam ErringtonLuca GiorgettiDave Penney,

Motivation : Understand nice inclusions A- CB of

von Neumann algebras with finite dimensional centres.

( Also understand representations of unitary multifusioncategories .)

Theorem [Popa'90) :

Finite depth finite index hyperfinite I , subfactors are

completely determined by the standard invariant.

Notation :

A- CB a unital inclusion of finite multifactors

A has minimal central projections p . . .. . . PaB has minimal central projections q . . .. .,q ,

Ai - Api , Bj = Bqj .

The inclusion is :

connected , if 2-(A) nZlB)E① iJones index

finite index , if whenever pigs. -40 , [Bqj : Ap;) 0 ;

isomorphic to IC15 if there is an isomorphism 4 :B→Iso that 41A)=A .

Connected and finite index will be standing assumptions .

Let tr be a faithful trace on B.

JA'TThe Jones basic construction \ ✓

A'

is JA'T cB( L4B, tr)) . B- L2B - B ' - JBJ

/ IJAJAs a von Neumann algebra A

it is generated by B and the projection e,:EB → EA .

Iterating we get the Jones tower A-AocB=A,cAzcAsc . - -

e? et es". .

tr is called Markov if there is a scalar µ> 0 and

an extension of tr to Az so that trlxe.) = µtrLx)

for all xeB . [GDHJ '893 : there is a unique Markov trace.

Jones formulation of the

The standard invariant is the system of relative commutants

(Afn An) , ( An AD which has the structure of a. (unitaryT

box spaces are

2- shaded) planar algebra . finite dimensional

E.g.s ftp..IE any eneajn.hn

f¥%f{ To AinAn A'on An . .

The planar algebra ( and the inclusion) is finite depth ifthere are finitely many equivalence classes of minimal projections .Equivalently, if sypdimZLA.tn An) a • .

Warning : Because we are not working with factors,

the 0 - box spaces are not 1 -dimensional .

We have

-W = ④ +④ + . . . +④

⑨④ + ③m%t . .+⑧

Downward basic construction of ACB :Markov index

projection eeB , E. Ale) - µ←

if C-- she}'nA then B isomorphic to the basic construction

for CCA .

If ACB factors,we can find a Jones tunnel :

- - - CA.sc A -z c A. , CA5A c. A.=D (Not unique !)et et et et e

.

"

Popa's result relies on finding such a tunnel .

What if there is no tunnel ?

Example :

(Ma'" ⑦ e)④ 12 does not admit a

C. ④ a downward basic construction

( Mile) ⑦ Ms④12

admits two steps ofdownward basic construction

Male ) Ot Mzce) but has isomorphic standardinvariant!

Iterating, we get countably many non- isomorphic inclusions

with the same standard invariants.

Theorem [ BCEGP ' 20] :

We have a bijection

¥:÷:÷÷÷¥±±ti¥⇒ .fi:÷÷:÷:÷:÷÷÷÷÷: .

A CB t ( PACB , trlzca))\ [ Markov trace

standard invariant

We'll focus on infectivity , i.e. , establishing that this is

a complete invariant .

Distortion

X a dualizable A,B - bimodule ( Think #⇐ IBD

Xij = piXqj an Ai ,Bj - bimodule

The modular distortion of X is

D= (di;) d.fi dimilailxijbdimpllx.jp;)defined for isj with Xij -70 .

Broad strokes :

- z a state on Po,+ determines distortion comments indifferentif + representations

- Given ACB there is Morita equivalent A'cB'

which admits a generating tunnel , so is classified

by [Dopa' 903

- Given IC15 with the same standard invariant,it is

Morita equivalent to Ects' EA'cB

'

- If ECB has the same distortion,these Morita equivalences

are unitary conjugate and so we get ACBE Acts .

More on distortion

X is extremal if- all A. Ai - and Bj ,Bj - bimodules generated by X

~ have equal left and right dimensionsI- all subbimodules of Xij have the same distortion

as A:B; - bimodules, andthere are 7ethao.GE/Rbso

So dij = 942; -

We extend 8 to all i. j inthis case .

Proposition :

Every finite depth connected dualizablen.kz is extremal .

ACB is extremal if AH3B is .

Notation :

4 the Jones dimension matrix

A = dimhl AH dim,zlHij)B;)X = ALZBB extremal

, dig. = 3%4. .

(we treat 3,2 as row vectors .)

Proposition :The distortion of for B- A,cA, is given by

Iii = IE,

did, = 4 i

4)j

i=HE'D ;

d. -is µ i

Immediate Consequence :If ACB admits a downward basic construction

,

its distortion must be of this form .

Drop : Write S = { Se Max, so)/ dij-3.sk}Then 5 → s

s → 13071) ;#i has a unique fixed points.

In fact, Jj = d×E where :

Xi

dx'

is the largest eigenvalue of DT0

xeIR% Be RIO unit vectors with

xD = d×p PDT - did

we call on the standard distortion.

Theorem : suppose ACB is a finite index inclusion of

finite multi factors . Then ACB admits a downward

basic construction if there is a projection IEB with

central support 1 so that

j,§ trjttjd.jo; = 1 for i. Is ... . a.

Theorem : suppose ACB is a finite index extremal connected

inclusion of finite multifactors . Then the inclusion is

homogeneous in the sense of [Popa'95] ldimfailp.IN) doesn't depend on :)

iff it has standard distortion .

( If A ,B are II., [Popa'953 shows these are equivalent

to admitting an infinite tunnel .)

Miel eX"= ¥T¥ ×,z=Io¥

/ I / X,= ×u=TI④

e e

• =L:D ⇐ I :?)

If it = tr§(7) e (0, 1)2

it' =L: +⇒ no solution

.

or = (Ii) ←n> 3 = ( 2 1) , 2=11 1)

I = (I 32) ← 3DT=(z 3) , 3=12 1)

£ = (%g %) arms ED'D :(

5 3) , EDT = (2 3)

1¥91 . ↳ %)

Morita EquivalenceActs ,

"a faithful right A- mod

Zi. Y '¥ LY3B

A'

= (A")'

NBLYA)n

B'

= (Bop)'nBlZB)

YA is a Morita equivalence of ACB and A'cB'

.

The induced isomorphism of standard invariants

is done by encircling box spaces .

Existence of Homogeneous Inclusions

Given a planar algebra , take an appropriate state on

the 0 - box spaces

Build a latticeu u

u

> 412 s ka s kiU u u

> 14 7 21 s

u u u

> 412 s 12 s Br

u u u

> 14 s k s

u v

Taking a"

leftward limit"

gives an inclusion with

the correct standard invariant and a tunnel .